We study transportation networks controlled by dynamic feedback tolls. We focus on a multiscale model, whereby the dynamics of the traffic flows are intertwined with those of the routing choices. The latter are influenced by the current traffic state of the network as well as by dynamic tolls controlled in feedback by the system planner. We prove that a class of decentralized monotone flow-dependent tolls allows for globally stabilizing the transportation network around a generalized Wardrop equilibrium. In particular, our results imply that using decentralized marginal cost tolls, stability of the dynamic transportation network is guaranteed around the social optimum traffic assignment. This is particularly remarkable as such dynamic feedback tolls can be computed in a fully local way without the need for any global information about the network structure, its state, or the exogenous network loads. Through numerical simulations, we also compare the performance of such decentralized dynamic feedback marginal cost tolls with constant offline (and centrally) optimized tolls both in the asymptotic and in the transient regime, and we investigate their robustness to information delays.

We study some optimal control problems on networks with junctions, approximate the junctions by a switching rule of delay-relay type, and study the passage to the limit when $\varepsilon$, the parameter of the approximation, goes to zero. First, for a twofold junction problem we characterize the limit value function as a viscosity solution and maximal subsolution of a suitable Hamilton--Jacobi problem. Then, for a threefold junction problem we consider two different approximations, recovering in both cases some uniqueness results in the sense of a maximal subsolution.